To solve the system of linear equations using the substitution method, we start with the given system:
[tex]\[
\begin{cases}
d + e = 2 \\
d - e = 4
\end{cases}
\][/tex]
### Step-by-Step Solution
1. Start by solving one of the equations for one of the variables.
We can solve the first equation for [tex]\(d\)[/tex]:
[tex]\[
d = 2 - e
\][/tex]
2. Substitute this expression into the other equation.
Substitute [tex]\(d = 2 - e\)[/tex] into the second equation:
[tex]\[
(2 - e) - e = 4
\][/tex]
Simplify the equation:
[tex]\[
2 - e - e = 4
\][/tex]
[tex]\[
2 - 2e = 4
\][/tex]
3. Solve for [tex]\(e\)[/tex].
Subtract 2 from both sides of the equation:
[tex]\[
-2e = 2
\][/tex]
Divide both sides by -2:
[tex]\[
e = -1
\][/tex]
4. Substitute [tex]\(e = -1\)[/tex] back into the equation [tex]\(d = 2 - e\)[/tex] to solve for [tex]\(d\)[/tex].
[tex]\[
d = 2 - (-1)
\][/tex]
Simplify:
[tex]\[
d = 2 + 1
\][/tex]
[tex]\[
d = 3
\][/tex]
5. Verify the solution with both original equations.
Substitute [tex]\(d = 3\)[/tex] and [tex]\(e = -1\)[/tex] back into the original equations to check:
For the first equation:
[tex]\[
3 + (-1) = 2
\][/tex]
This is true since [tex]\(2 = 2\)[/tex].
For the second equation:
[tex]\[
3 - (-1) = 4
\][/tex]
This is also true since [tex]\(4 = 4\)[/tex].
Therefore, the solution to the system is:
[tex]\[
(d, e) = (3, -1)
\][/tex]
So, the correct choice is: The solution is [tex]\((3, -1)\)[/tex].
